research-article
Authors: Akbar Shirilord and Mehdi Dehghan
Published: 25 June 2024 Publication History
- 0citation
- 0
- Downloads
Metrics
Total Citations0Total Downloads0Last 12 Months0
Last 6 weeks0
New Citation Alert added!
This alert has been successfully added and will be sent to:
You will be notified whenever a record that you have chosen has been cited.
To manage your alert preferences, click on the button below.
Manage my Alerts
New Citation Alert!
Please log in to your account
- View Options
- References
- Media
- Tables
- Share
Abstract
This study presents some new iterative algorithms based on the gradient method to solve general constrained systems of conjugate transpose matrix equations for both real and complex matrices. In addition, we analyze the convergence properties of these methods and provide numerical techniques to determine the solutions. The effectiveness of the proposed iterative methods is demonstrated through various numerical examples employed in this study.
References
[1]
R. Ali, K. Pan, The new iteration methods for solving absolute value equations, Appl. Math. 68 (2023) 109–122.
[2]
Z.-Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003) 603–626.
Digital Library
[3]
Z.-Z. Bai, The convergence of the two-stage iterative method for hermitian positive definite linear systems, Appl. Math. Lett. 11 (1998) 1–5.
[4]
Z.-Z. Bai, G.H. Golub, J.-Y. Pan, Preconditioned Hermitian and skew- Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math. 98 (2004) 1–32.
[5]
P. Chansangiam, Closed forms of general solutions for rectangular systems of coupled generalized Sylvester matrix differential equations, Commun. Math. Appl. 11 (2020) 311–324.
[6]
H.C. Chen, Generalized reflexive matrices: special properties and applications, SIAM J. Matrix Anal. Appl. 19 (1998) 140–153.
Digital Library
[7]
O.L.V. Costa, M.D. Fragoso, Stability results for discrete-time linear systems with Markovian jumping parameters, J. Math. Anal. Appl. 179 (1993) 154–178.
[8]
M. Dehghan, A. Shirilord, Accelerated double-step scale splitting iteration method for solving a class of complex symmetric linear systems, Numer. Algorithms 83 (2020) 281–304.
[9]
M. Dehghan, M. Hajarian, Solving the system of generalized Sylvester matrix equations over the generalized centro-symmetric matrices, J. Vib. Control 20 (2014) 838–846.
[10]
M. Dehghan, M. Hajarian, SSHI methods for solving general linear matrix equations, Eng. Comput. 28 (2011) 1028–1043.
[11]
M. Dehghan, M. Hajarian, Construction of an iterative method for solving generalized coupled Sylvester matrix equations, Trans. Inst. Meas. Control 35 (2013) 961–970.
[12]
M. Dehghan, R. Mohammadi-Arani, Generalized product-type methods based on bi-conjugate gradient (GPBiCG) for solving shifted linear systems, Comput. Appl. Math. 36 (2017) 1591–1606.
[13]
M. Dehghan, A. Shirilord, A new approximation algorithm for solving generalized Lyapunov matrix equations, J. Comput. Appl. Math. 404 (2022).
[14]
J. Ding, Y.-J. Liu, F. Ding, Iterative solutions to matrix equations of the form A i X B i = F i, Comput. Math. Appl. 59 (2010) 3500–3507.
[15]
F. Ding, P.X. Liu, J. Ding, Iterative solutions of the generalized Sylvester matrix equation by using the hierarchical identification principle, Appl. Math. Comput. 197 (2008) 41–50.
[16]
F. Ding, H.M. Zhang, Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems, IET Control Theory Appl. 8 (2014) 1588–1595.
[17]
F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim. 44 (2006) 2269–2284.
Digital Library
[18]
F. Ding, Least squares parameter estimation and multi-innovation least squares methods for linear fitting problems from noisy data, J. Comput. Appl. Math. 426 (2023).
[19]
F. Ding, L. Xu, X. Zhang, Y. Zhou, Filtered auxiliary model recursive generalized extended parameter estimation methods for Box–Jenkins systems by means of the filtering identification idea, Int. J. Robust Nonlinear Control 33 (2023) 5510–5535.
[20]
B.D. Djordjević, Singular Sylvester equation in Banach spaces and its applications: Fredholm theory approach, Linear Algebra Appl. 622 (2021) 189–214.
[21]
B.D. Djordjević, N.Č Dinčić, Classification and approximation of solutions to Sylvester matrix equation, Filomat 33 (2019) 4261–4280.
[22]
M. Donatelli, C. Garoni, M. Mazza, S. Serra-Capizzano, D. Sesana, Preconditioned HSS method for large multilevel block Toeplitz linear systems via the notion of matrix-valued symbol, Numer. Linear Algebra Appl. 23 (2016) 83–119.
[23]
H. Dorissen, Canonical forms for bilinear systems, Syst. Control Lett. 13 (1989) 153–160.
[24]
X.-B. Feng, K.A. Loparo, Y.-D. Ji, H.J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Autom. Control 37 (1992) 38–53.
[25]
K. Glover, D.J.N. Limebeer, J.C. Doyl, E.M. Kasenally, M.G. Safonov, A characterisation of all solutions to the four block general distance problem, SIAM J. Control Optim. 29 (1991) 283–324.
[26]
M. Hajarian, Solving the general Sylvester discrete-time periodic matrix equations via the gradient based iterative method, Appl. Math. Lett. 52 (2016) 87–95.
[27]
M. Hajarian, Generalized conjugate direction algorithm for solving the general coupled matrix equations over symmetric matrices, Numer. Algorithms 73 (2016) 591–609.
[28]
M. Hajarian, Computing symmetric solutions of general Sylvester matrix equations via Lanczos version of biconjugate residual algorithm, Comput. Math. Appl. 76 (2018) 686–700.
[29]
Z.H. He, O.M. Agudelo, Q.W. Wang, B. De Moor, Two-sided coupled generalized Sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices, Linear Algebra Appl. 496 (2016) 549–593.
[30]
L.B. Iantovics, F.F. Nichita, On the colored and the set-theoretical Yang–Baxter equations, Axioms 10 (2021) 146,.
[31]
T. Jiang, M. Wei, On solutions of the matrix equations X − A X B = C and X − A X ‾ B = C, Linear Algebra Appl. 367 (2003) 225–233.
[32]
K. Jbilou, A.J. Riquet, Projection methods for large Lyapunov matrix equations, Linear Algebra Appl. 415 (2006) 344–358.
[33]
Y.-F. Ke, C.-F. Ma, Alternating direction method for generalized Sylvester matrix equation A X B + C Y D = E, Appl. Math. Comput. 260 (2015) 106–125.
[34]
Y.F. Ke, C.F. Ma, The alternating direction methods for solving the Sylvester-type matrix equation A X B + C X T D = E, J. Comput. Math. 35 (2017) 620–641.
[35]
I.I. Kyrchei, D. Mosić, P.S. Stanimirović, MPD-DMP-solutions to quaternion two-sided restricted matrix equations, Comput. Appl. Math. 40 (2021) 177.
[36]
Z.Y. Li, Y. Wang, B. Zhou, G.R. Duan, Least squares solution with the minimum-norm to general matrix equations via iteration, Appl. Math. Comput. 215 (2010) 3547–3562.
[37]
T.-T. Lu, S.-H. Shiou, Inverses of 2 × 2 block matrices, Comput. Math. Appl. 43 (2002) 119–129.
[38]
B.C. Moore, Principal component analysis in linear systems: controllability, observability and model reduction, IEEE Trans. Autom. Control 26 (1981) 17–31.
[39]
F.F. Nichita, Mathematics and Poetry • Unification, Unity, Union, Sci 2 (4) (2020).
[40]
Q. Niu, Y. Wang, L.-Z. Lu, A relaxed gradient based algorithm for solving Sylvester equations, Asian J. Control 13 (2011) 461–464.
[41]
D.A. Paolo, I. Alberto, R. Antonio, Realization and structure theory of bilinear dynamical systems, SIAM J. Control Optim. 12 (1974) 517–535.
[42]
X. Sheng, A relaxed gradient based algorithm for solving generalized coupled Sylvester matrix equations, J. Franklin Inst. 355 (2018) 4282–4297.
[43]
A. Shirilord, M. Dehghan, Closed-form solution of non-symmetric algebraic Riccati matrix equation, 131 (2022) 108040.
[44]
P.S. Stanimirović, M.D. Petković, Gradient neural dynamics for solving matrix equations and their applications, Neurocomputing 306 (2018) 200–212.
[45]
Y. Tang, J. Peng, S. Yue, Cyclic and simultaneous iterative methods to matrix equations of the form A i Y B i = F i, Numer. Algorithms 66 (2014) 379–397.
[46]
W. Wang, C. Song, A novel iterative method for solving the coupled Sylvester-conjugate matrix equations and its application in antilinear system, J. Appl. Anal. Comput. 1 (2023) 249–274.
[47]
Q.W. Wang, A. Rehman, Z.H. He, Y. Zhang, Constraint generalized Sylvester matrix equations, Automatica 69 (2016) 60–64.
[48]
Y. Wang, F. Ding, Novel data filtering based parameter identification for multiple-input multiple-output systems using the auxiliary model, Automatica 71 (2016) 308–313.
[49]
W. Wang, C. Song, Iterative algorithms for discrete-time periodic Sylvester matrix equations and its application in antilinear periodic system, Appl. Numer. Math. 168 (2021) 251–273.
[50]
B. Kágström, L. Westin, Generalized Schur methods with condition estimators for solving the generalized Sylvester equation, IEEE Trans. Autom. Control 34 (1989) 745–751.
[51]
A.-G. Wu, G. Feng, G.-R. Duan, W.-J. Wu, Iterative solutions to coupled Sylvester-conjugate matrix equations, Comput. Math. Appl. 60 (2010) 54–66.
[52]
A.-G. Wu, G. Feng, G.-R. Duan, W.-J. Wu, Closed-form solutions to Sylvester-conjugate matrix equations, Comput. Math. Appl. 60 (2010) 95–111.
[53]
A.G. Wu, G. Feng, J.Q. Hu, G.R. Duan, Closed-form solutions to the nonhom*ogeneous Yakubovich-conjugate matrix equation, Appl. Math. Comput. 214 (2009) 442–450.
[54]
A.G. Wu, G.R. Duan, Solution to the generalised Sylvester matrix equation A V + B W = E V F, IET Control Theory Appl. 1 (2007) 402–408.
[55]
Y.-J. Xie, N. Huang, C.-F. Ma, Iterative method to solve the generalized coupled Sylvester-conjugate transpose linear matrix equations over reflexive or anti-reflexive matrix, Comput. Math. Appl. 67 (2014) 2071–2084.
[56]
L. Xie, J. Ding, F. Ding, Gradient based iterative solutions for general linear matrix equations, Comput. Math. Appl. 58 (2009) 1441–1448.
[57]
L. Xie, Y.J. Liu, H.Z. Yang, Gradient based and least squares based iterative algorithms for matrix equations A X B + C X T D = F, Appl. Math. Comput. 217 (2010) 2191–2199.
[58]
L. Xu, Separable Newton recursive estimation method through system responses based on dynamically discrete measurements with increasing data length, Int. J. Control. Autom. Syst. 20 (2022) 432–443.
[59]
L. Xu, Parameter estimation for nonlinear functions related to system responses, Int. J. Control. Autom. Syst. 21 (2023) 1780–1792.
[60]
L. Xu, F. Ding, Q. Zhu, Separable synchronous multi-innovation gradient-based iterative signal modeling from on-line measurements, IEEE Trans. Instrum. Meas. 71 (2022) 1–13.
[61]
Y.-J. Yie, C.-F. Ma, The MGPBiCG method for solving the generalized coupled Sylvester-conjugate matrix equations, Appl. Math. Comput. 265 (2015) 68–78.
[62]
H. Zhang, H. Yin, New proof of the gradient-based iterative algorithm for the Sylvester conjugate matrix equation, Comput. Math. Appl. 74 (2017) 3260–3270.
[63]
H.M. Zhang, Quasi gradient-based inversion-free iterative algorithm for solving a class of the nonlinear matrix equations, Comput. Math. Appl. 77 (2019) 1233–1244.
Digital Library
[64]
H.M. Zhang, H.C. Yin, Refinements of the Hadamard and Cauchy-Schwarz inequalities with two inequalities of the principal angles, J. Math. Inequal. 13 (2019) 423–435.
[65]
X. Zhang, F. Ding, Optimal adaptive filtering algorithm by using the fractional-order derivative, IEEE Signal Process. Lett. 29 (2021) 399–403.
[66]
Y. Zhou, X. Zhang, F. Ding, Partially-coupled nonlinear parameter optimization algorithm for a class of multivariate hybrid models, Appl. Math. Comput. 414 (2022).
Digital Library
[67]
B. Zhou, J. Lam, G.-R. Duan, Gradient-based maximal convergence rate iterative method for solving linear matrix equations, Int. J. Comput. Math. 87 (2010) 515–527.
Digital Library
[68]
B. Zhou, J. Lam, G.-R. Duan, Toward solution of matrix equation X = A f ( X ) B + C, Linear Algebra Appl. 435 (2011) 1370–1398.
Recommendations
- The generalised Sylvester matrix equations over the generalised bisymmetric and skew-symmetric matrices
A matrix <italic>P</italic> is called a symmetric orthogonal if <italic>P</italic> = <italic>P</italic><italic>T</italic> = <italic>P</italic>−1. A matrix <italic>X</italic> is said to be a generalised bisymmetric with respect to <italic>P</italic> if <...
Read More
- Convergence of accelerated modulus-based matrix splitting iteration methods for linear complementarity problem with an H + -matrix
The theoretical analysis of the accelerated modulus-based matrix splitting iteration methods for the solution of the large sparse linear complementarity problem is further studied. The convergence conditions are presented by fully utilizing the H + -...
Read More
- Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem
For the large sparse linear complementarity problem, a class of accelerated modulus-based matrix splitting iteration methods is established by reformulating it as a general implicit fixed-point equation, which covers the known modulus-based matrix ...
Read More
Comments
Information & Contributors
Information
Published In
Applied Numerical Mathematics Volume 198, Issue C
Apr 2024
508 pages
ISSN:0168-9274
Issue’s Table of Contents
IMACS.
Publisher
Elsevier Science Publishers B. V.
Netherlands
Publication History
Published: 25 June 2024
Author Tags
- Iterative method
- Convergence
- Conjugate transpose matrix equations
- Image restoration
- Markovian jump systems
- Linear time-invariant dynamical systems
Qualifiers
- Research-article
Contributors
Other Metrics
View Article Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
Total Citations
Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Other Metrics
View Author Metrics
Citations
View Options
View options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in
Full Access
Get this Publication
Media
Figures
Other
Tables
